Absolute ValueDefinition, How to Discover Absolute Value, Examples
Many comprehend absolute value as the distance from zero to a number line. And that's not incorrect, but it's nowhere chose to the complete story.
In mathematics, an absolute value is the magnitude of a real number without considering its sign. So the absolute value is always a positive number or zero (0). Let's check at what absolute value is, how to discover absolute value, some examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a number is constantly zero (0) or positive. It is the extent of a real number without regard to its sign. That means if you possess a negative figure, the absolute value of that figure is the number ignoring the negative sign.
Meaning of Absolute Value
The last definition states that the absolute value is the length of a figure from zero on a number line. So, if you think about it, the absolute value is the length or distance a figure has from zero. You can visualize it if you take a look at a real number line:
As demonstrated, the absolute value of a figure is the distance of the number is from zero on the number line. The absolute value of negative five is five reason being it is five units apart from zero on the number line.
Examples
If we graph negative three on a line, we can observe that it is 3 units apart from zero:
The absolute value of negative three is 3.
Now, let's look at another absolute value example. Let's say we hold an absolute value of 6. We can graph this on a number line as well:
The absolute value of 6 is 6. So, what does this refer to? It states that absolute value is at all times positive, regardless if the number itself is negative.
How to Find the Absolute Value of a Number or Expression
You need to know a couple of points before going into how to do it. A couple of closely related features will support you grasp how the number within the absolute value symbol works. Thankfully, what we have here is an meaning of the following four rudimental features of absolute value.
Basic Characteristics of Absolute Values
Non-negativity: The absolute value of ever real number is constantly zero (0) or positive.
Identity: The absolute value of a positive number is the expression itself. Instead, the absolute value of a negative number is the non-negative value of that same expression.
Addition: The absolute value of a total is less than or equal to the sum of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With these four basic properties in mind, let's look at two more helpful characteristics of the absolute value:
Positive definiteness: The absolute value of any real number is always positive or zero (0).
Triangle inequality: The absolute value of the variance among two real numbers is lower than or equivalent to the absolute value of the total of their absolute values.
Considering that we learned these properties, we can ultimately start learning how to do it!
Steps to Calculate the Absolute Value of a Figure
You need to follow a couple of steps to calculate the absolute value. These steps are:
Step 1: Write down the number whose absolute value you want to discover.
Step 2: If the expression is negative, multiply it by -1. This will change it to a positive number.
Step3: If the figure is positive, do not convert it.
Step 4: Apply all characteristics applicable to the absolute value equations.
Step 5: The absolute value of the number is the expression you have following steps 2, 3 or 4.
Bear in mind that the absolute value symbol is two vertical bars on either side of a figure or expression, similar to this: |x|.
Example 1
To begin with, let's assume an absolute value equation, like |x + 5| = 20. As we can see, there are two real numbers and a variable inside. To solve this, we are required to calculate the absolute value of the two numbers in the inequality. We can do this by following the steps mentioned priorly:
Step 1: We are given the equation |x+5| = 20, and we have to calculate the absolute value inside the equation to find x.
Step 2: By using the fundamental properties, we learn that the absolute value of the total of these two numbers is equivalent to the total of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unknown, so let's remove the vertical bars: x+5 = 20
Step 4: Let's solve for x: x = 20-5, x = 15
As we can observe, x equals 15, so its distance from zero will also be as same as 15, and the equation above is genuine.
Example 2
Now let's work on another absolute value example. We'll use the absolute value function to solve a new equation, such as |x*3| = 6. To do this, we again have to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We have to solve for x, so we'll start by dividing 3 from each side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two possible results: x = 2 and x = -2.
Step 4: Therefore, the initial equation |x*3| = 6 also has two likely results, x=2 and x=-2.
Absolute value can involve a lot of complex expressions or rational numbers in mathematical settings; nevertheless, that is something we will work on separately to this.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is differentiable at any given point. The following formula provides the derivative of the absolute value function:
f'(x)=|x|/x
For absolute value functions, the domain is all real numbers except zero (0), and the range is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function is constant at 0, so the derivative of the absolute value at 0 is 0.
The absolute value function is not differentiable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is given by:
I'm →0−(|x|/x)
The right-hand limit is given by:
I'm →0+(|x|/x)
Since the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinctable at zero (0).
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